SIR Model for Portugal - North Region

Last Updated: 2020-04-19 12:20:20 UTC

1 Disclaimer

This is an independent simulation and does not reflect any portuguese governmental position about the subject.

This is a work in progress.

2 Concepts

• Incubation Period: is the time between infection and the visible appearance of symptoms.
• Latency period: is the period of time between infection and the ability to infect someone.

3 Model Hypothesis

3.1 Compartments

The population is assumed to be divided in compartments, for example:

• Susceptible individuals
• Infected, symptomatic and infectious
• Recovered, immune from further infection

3.2 Discrete time

The observations, for simplicity, will be analyzed as a discrete time-series. In this case we will use days.

3.3 Population

The total number of individuals is modeled using currently available data for birth and death rates.

3.4 Other assumptions

• The population is considered big, so the random effects can be ignored.
• The population is considered homogeneous at any point in time. This means that the individuals of any compartment are distributed randomically.
• The disease is transmitted by proximity or contact between an infected and a susceptible individual (individual contact model - ICM).
• The susceptible individual becomes infected instantaneously.
• The infected individuals may or may not be infectious, until they get symptoms.
• The infected eventually get recovered and become immune, or die.
• The population pyramid is taking in account for disease severity.

4 Extended Model

A more complex model is proposed by Tim Churches¹, adding several compartments and permit a better exploration of the potential effects of various combinations and timings of interventions on COVID-19 spread.

Attention to the description of each compartment:

Compartment	Functional definition
S	Susceptible individuals
E	Exposed and infected, not yet symptomatic but potentially infectious
I	Infected, symptomatic and infectious
Q	Infectious, but (self-)isolated
H	Requiring hospitalisation (would normally be hospitalised if capacity available)
R	Recovered, immune from further infection
F	Case fatality (death due to COVID-19, not other causes)

4.1 Transition diagram

4.2 Parameters

There is a lot of parameters. We will start with the defaults and adjust accordingly to new evidences.

The full list of parameters is available in the source code.

4.3 Time-variant parameters

Several of the simulation parameters can also be varied over time. That is, as well as accepting a single value, they also accept a vector of values with length equal to nsteps, the number of days over which the simulation runs. Assuming nsteps=366, we can, for example, set quar.rate = c(rep(1/10,30), rep(1/3, 335)). That defines a step function in which the transition-to-isolation rate for the I compartment is 0.1 for the first 30 days, then steps up to 0.33 thereafter, reflecting, say, a campaign or government edict to self-isolate. Of course, the vector of values can be a smooth function, or can go up then down again, or whatever you wish to model. This is an extremely powerful feature of computational models such as this, and one very hard to incorporate into purely mathematical models.

Most of the parameters for which time-variation is useful support it. If they don’t, you will receive an error message (the table above will be updated to include this information later).

5 Evidence

5.1 COVID-related

Last checked: 2020-04-11

• Recovery (h, d) = takes at least 15 days
• Exposition to symptoms (b) = median of 5
• Incubation period: 5.1 (2.2 - 11.5)²
• Symptoms to death: ~15 (11 - 17) (So, Infection to death ~20 days)²
• Days to double: 8-9 (Case report estimation is 31-48% at 2020-04-11, for R 1.3 and CFR 2.0³)
• Mortality rate: 0.85 - 1.31%³

6 Baseline simulation

6.1 Parameters used

• Population: 3,574,394 (estimation of 2019 from Census)⁴
• Birth rate: 7.7 ‰⁴
• Baseline death rate (any cause): 9.86 ‰⁵
• Baseline hospital death rate (any cause): 50.5 ‰⁵
• Rates of exposition (table below)
• Report rate (table below)
• Ages and severity (table below)

Interaction rates between compartments
Date	Event	E-S	I-S	Q-S
08 March	Suspension of visits to health facilities and prisons	16	10	5.0
10 March	Suspension of flights from the most affected areas of Italy	15	10	5.0
11 March	WHO declares pandemic	15	10	5.0
13 March	Portugal alert status statement	10	7	3.5
15 March	Restrictions on access to commercial spaces	7	5	2.5
16 March	Suspension of classes	5	3	1.5
18 March	Declaration of state of emergency	5	2	1.0
02 April	Extension of the state of emergency	3	2	1.0

Age piramid and disease fatality rate
Ages	Prevalence	Fatality rate
>=0	8.03%	1.30%
>=10	10.34%	1.30%
>=20	11.16%	1.30%
>=30	12.35%	1.30%
>=40	15.51%	1.30%
>=50	15.36%	1.30%
>=60	12.76%	2.60%
>=70	8.68%	3.90%
>=80	5.79%	5.20%

6.2 Distribution of duration in key compartments

6.3 Baseline Prevalence

6.3.1 Overall - Interactive Plot

6.3.2 Comparison with real data - Interactive Plot

6.3.3 Basic reproduction number \(R_{0}\)

Estimated R is 1.2 (±0.02)

7 Running an intervention experiments

Work in progress…

8 Acknowledgments

This simulations are heavily based on Tim Churches’ code, available here.

1. Churches T. Tim churches health data science blog: Modelling the effects of public health interventions on covid-19 transmission using r - part 2. 2020. https://timchurches.github.io/blog/posts/2020-03-18-modelling-the-effects-of-public-health-interventions-on-covid-19-transmission-part-2/.

2. MIDAS-Network. COVID-19 parameter estimates. https://github.com/midas-network/COVID-19/tree/master/parameter_estimates/2019_novel_coronavirus.

3. CMMID. Inferring covid-19 cases from deaths of confirmed cases. https://cmmid.github.io/visualisations/inferring-covid19-cases-from-deaths.

4. Instituto nacional de estatística. https://www.ine.pt.

5. Transparência sns. https://transparencia.sns.gov.pt.